L(s) = 1 | − 3·3-s + 8.16·5-s + 9·9-s − 37.8·11-s + 39.9·13-s − 24.5·15-s + 9.93·17-s + 90.4·19-s − 118.·23-s − 58.2·25-s − 27·27-s − 78.4·29-s − 92.0·31-s + 113.·33-s + 332.·37-s − 119.·39-s + 71.7·41-s + 115.·43-s + 73.5·45-s − 307.·47-s − 29.8·51-s − 403.·53-s − 308.·55-s − 271.·57-s − 593.·59-s + 333.·61-s + 326.·65-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.730·5-s + 0.333·9-s − 1.03·11-s + 0.851·13-s − 0.421·15-s + 0.141·17-s + 1.09·19-s − 1.07·23-s − 0.466·25-s − 0.192·27-s − 0.502·29-s − 0.533·31-s + 0.598·33-s + 1.47·37-s − 0.491·39-s + 0.273·41-s + 0.411·43-s + 0.243·45-s − 0.955·47-s − 0.0818·51-s − 1.04·53-s − 0.757·55-s − 0.630·57-s − 1.31·59-s + 0.699·61-s + 0.622·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 8.16T + 125T^{2} \) |
| 11 | \( 1 + 37.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 39.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 9.93T + 4.91e3T^{2} \) |
| 19 | \( 1 - 90.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 118.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 78.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 92.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 332.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 71.7T + 6.89e4T^{2} \) |
| 43 | \( 1 - 115.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 307.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 403.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 593.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 333.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 743.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 728.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 801.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.06e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 906.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.11e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.48e3T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.045330487103058900905664341959, −7.60984289519692954290088459763, −6.48334105181682943704770685926, −5.80363985078780630500261373229, −5.34399317786656856517728070405, −4.30643098317572533207798804575, −3.28286569971971445905894587307, −2.19615861395207434311345401107, −1.22566106239991888515509576555, 0,
1.22566106239991888515509576555, 2.19615861395207434311345401107, 3.28286569971971445905894587307, 4.30643098317572533207798804575, 5.34399317786656856517728070405, 5.80363985078780630500261373229, 6.48334105181682943704770685926, 7.60984289519692954290088459763, 8.045330487103058900905664341959